I occasionally have this issue with my iphone 6 (and my iphone 5 before that). If you swipe up from the bottom to get into control center, on the lower right is where the audio is going. It will say "Volt" but nothing is playing. Press that and switch it to "iphone" and the sound will start coming back out of the phone's speaker. Then switch it back to "Volt" and it's worked 100% of the time for me. Takes just a few seconds. May be faster than your current method? Worth a try...I have this problem of Bluetooth pairing and no audio all the time. The trick for me to circumvent the problem is to be patient. When you start the car, don't touch the radio until at least 45 seconds, then even wait a some more. Then when you switch the audio source to Bluetooth the audio will flow. I can recreate the problem every time with my iPhone SE. If I pair and try to play music too quickly, the display shows it playing, the phone thinks it's playing, but no audio. Once in this mode, the only way around it is to turn off Bluetooth on the phone, wait, turn it back on, wait, wait some more, then switch the source to Bluetooth and play.
The other thing to check is if your audio on the phone might be turned all the way down. I sometimes need to push my volume up buttons on the phone.
There's always the trusted analog headphone cable...that always works for music, not so much for phone features.
OP... bluetooth is supposed to be standard, so if both devices claim to be bluetooth, they are supposed to work together. For some reason, car and phone makers (not sure who is more to blame) sometimes implement the standard in a non-standard way and it can cause issues. See my comment above, perhaps your Pixel is doing something similar, can you switch between Volt and phone speaker output and back and see if that fixes it? Did you make sure the volume is not at 0 on the phone? Have you verified the pixel works with other bluetooth devices (speakers, headsets, etc)?